Many references (see for
example [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]})
provide criteria allowing to characterize the asymptotic behavior of the semigroup \((P_n)_{n\in \mathbb {Z}_+}\)  in the following form: there
exist a measurable function \(V:E\rightarrow [1,+\infty )\) , constants \(\theta _0\in (0,1]\) , \(\alpha \in (0,1)\)  and \(C\in \mathbb {R}_+\) , a
probability measure \(\nu _\text{QS}\)  on \(E\)  such that \(\nu _\text{QS}(V)<+\infty \)  and a measurable function \(\eta :E\rightarrow \mathbb {R}_+\) 
non-identically zero such that \(\eta /V\)  is bounded and \(\nu _\text{QS}(\eta )>0\) , such that
\(\left|\theta _0^{-n} P_n f(x)-\eta (x)\nu _\text{QS}(f)\right|\le C\alpha ^n V(x)\) 
